Supports of certain infinitely divisible probability measures on locally convex spaces. (English) Zbl 0776.60008

Summary: Let \(B\) be a separable Banach space and let \(\mu\) be a centered Poisson probability measure on \(B\) with Lévy measure \(M\). Assume that \(M\) admits a polar decomposition in terms of a finite measure \(\sigma\) on the unit sphere of \(B\) and a Lévy measure \(\rho\) on \((0,\infty)\). The main result of this paper provides a complete description of the structure of \({\mathcal S}_ \mu\), the support of \(\mu\). Specifically, it is shown that: (i) if \(\int_{(0,1]}s \rho(ds)=\infty\), then \({\mathcal S}_ \mu\) is a linear space and is equal to the closure of the semigroup generated by \({\mathcal S}_ M\) (the support of \(M)\) and the negative of the barycenter of \(\sigma\); and (ii) if \(\int_{(0,1]}s \rho(ds)<\infty\) and zero is in the support of \(\rho\), then \({\mathcal S}_ \mu\) is a convex cone and is equal to the closure of the semigroup generated by \({\mathcal S}_ M\). The result (i) yields an affirmative answer to the question, open for some time, of whether the support of a stable probability measure of index \(1 \leq \alpha<2\) on \(B\) is a translate of a linear space. Analogs of these results, for both Poisson and stable probability measures defined on general locally convex spaces, are also provided.


60B11 Probability theory on linear topological spaces
60E07 Infinitely divisible distributions; stable distributions
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